Question

1. A sample of gaseous uranium hexafluoride, UF6, is held at a temperature of 300 K and a pressure of 0.1 mbar. The collision diameter of UF6 is 0.40 nm. (i) Calculate the root-mean-square (r.m.s.) speed of the molecules. Make sure to show all units in your working and how the units cancel out. [5 marks] (ii) Estimate the mean free path and collision frequency under these conditions. [10 marks]

Answer 1

$V_{rms}=\sqrt{\frac{3RT}{M}}$

Molar mass of $UF_6=352g/mol$

$\therefore\>M=0. 352kg/mol$

$R=8.314J/mol /K$

$T=300K$

$V_{rms}=\frac{\sqrt{3×8.314Jmol^{-1}K^{-1}300K}}{0.352kgmol^{-1}}$

$=145.8\sqrt{Jkg^{-1}}$

But $J=kgm^2s^{-2}$

$=145.8\sqrt{kgm^2s^{-2}kg^{-1}}$

$=145.8m/s$

Part (ii)

Mean free path $\lambda$ is given by

$\lambda=\frac{KT}{\sqrt{2}\pi\>d^2P}$

$=\frac{1.38×10^{-23}×300}{\sqrt{2}×\pi×(0.4×10^{-9})^2×0.1×10^2}$

$=5.824×10^{-4}m$

Collision frequency

$=\frac{V_{rms}}{\lambda}=\frac{145.8}{5.824×10^{-4}}=2.503×10^5s^{-1}$